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Transcript of Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.7 Negative Exponents and...
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Section 5.7
NegativeExponents and
ScientificNotation
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 2
Objective #1 Use the negative exponent rule.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 3
If b is any real number other than 0 and n is a natural number, then
When a negative number appears as an exponent, switch the position of the base (from numerator to denominator or denominator to numerator) and make the exponent positive. The sign of the base does not change.
1 1 and .n n
n nb b
b b
Negative Exponents in Numerators and Denominators
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4
4
44
3
33
Write with positive exponents.
1
Write 5 with positive exponents.
15
5
x
xx
Negative Exponents
EXAMPLESEXAMPLES
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5
4
44
3
33
Write with positive exponents.
1
Write 5 with positive exponents.
15
5
x
xx
Negative Exponents
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Objective #1: Example
1a. Use the negative exponent rule to write 35 with a positive exponent and then simplify.
33
15
51
125
1b. Use the negative exponent rule to write 4( 3) with a positive exponent and then simplify.
44
1( 3)
( 3)
181
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Objective #1: Example
1a. Use the negative exponent rule to write 35 with a positive exponent and then simplify.
33
15
51
125
1b. Use the negative exponent rule to write 4( 3) with a positive exponent and then simplify.
44
1( 3)
( 3)
181
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8
Objective #1: Example
1c. Use the negative exponent rule to write 18 with a positive exponent and then simplify.
111
8818
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Objective #1: Example
1c. Use the negative exponent rule to write 18 with a positive exponent and then simplify.
111
8818
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10
Objective #1: Example
1d. Write 24
5
with positive exponents only.
Then simplify, if possible.
2 2
24 55 4
2516
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11
Objective #1: Example
1d. Write 24
5
with positive exponents only.
Then simplify, if possible.
2 2
24 55 4
2516
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12
Objective #1: Example
1e. Write 2
1
7 y with positive exponents only.
Then simplify, if possible.
2
21
77
y
y
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13
Objective #1: Example
1e. Write 2
1
7 y with positive exponents only.
Then simplify, if possible.
2
21
77
y
y
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14
1f. Write 1
8x
y
with positive exponents only.
Then simplify, if possible.
1 8
8 1
8
x y
y x
yx
Objective #1: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15
1f. Write 1
8x
y
with positive exponents only.
Then simplify, if possible.
1 8
8 1
8
x y
y x
yx
Objective #1: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16
Objective #2 Simplify exponential expressions.
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Simplifying Exponential Expressions
An exponential expression is “simplified” when
a. Each base occurs only once.
b. No parentheses appear.
c. No powers are raised to powers.
d. No negative or zero exponents appear.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18
Simplifying Exponential Expressions
4444 22 baab
134919711971 QQQ
632
3
32
33
2
4913171717
xxxx
50105105 WWW
Simplification Techniques
Examples
If necessary, remove parentheses by using the Products to Powers Rule or the Quotient to Powers Rule.
If necessary, simplify powers to powers by using the Power Rule.
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Simplifying Exponential Expressions
20164164 HHHH
51234523043 YX
6172317
23
VVV
V
1212
3131
KK
Simplification Techniques
Examples
Be sure each base appears only once in the final form by using the Product Rule or Quotient Rule
If necessary, rewrite exponential expressions with zero powers as 1. Furthermore, write the answer with positive exponents by using the Negative Exponent Rule
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20
Simplify: 834
5
)2(
x
x
When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.2
2
8
6
8
323
5
85
8
5
8
5
)(2
x
x
x
x
x
x
Cube each factor in the numerator.
Multiply powers using (bm)n = bmn.
Division with the same base, subtract exponents.
Simplifying Exponential Expressions
EXAMPLEEXAMPLE
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Simplify: 2 3
8
(2 )
5
x
x
When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.
3 2 3
8
6
8
2
2
2 ( )
5
8
5
8
58
5
x
x
x
x
x
x
Cube each factor in the numerator.
Multiply powers using (bm)n = bmn.
Division with the same base, subtract exponents.
Simplifying Exponential Expressions
EXAMPLEEXAMPLE
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2a. Simplify: 12 2x x
12 2 12 2
10
101
x x x
x
x
Objective #2: Example
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2a. Simplify: 12 2x x
12 2 12 2
10
101
x x x
x
x
Objective #2: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24
Objective #2: Example
2b. Simplify: 3
975
5
x
x
3 3
9 9
3 9
6
6
75 7555
15
15
15
x x
x x
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25
Objective #2: Example
2b. Simplify: 3
975
5
x
x
3 3
9 9
3 9
6
6
75 7555
15
15
15
x x
x x
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26
Objective #2: Example
2c. Simplify: 4 2
11(6 )x
x
4 2 2 4 2
11 11
4 2
11
8
11
8 11
3
3
(6 ) 6 ( )
36
36
36
36
36
x x
x x
x
x
x
x
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27
Objective #2: Example
2c. Simplify: 4 2
11(6 )x
x
4 2 2 4 2
11 11
4 2
11
8
11
8 11
3
3
(6 ) 6 ( )
36
36
36
36
36
x x
x x
x
x
x
x
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28
Objective #2: Example
2d. Simplify:
58
4x
x
58 54
4
20
201
xx
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 29
Objective #2: Example
2d. Simplify:
58
4x
x
58 54
4
20
201
xx
x
x
x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30
Objective #3 Convert from scientific notation to
decimal notation.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31
Scientific Notation
At times you may find it necessary to work with really large numbers, or alternately, really small numbers. Here you will learn how to write these often cumbersome numbers in scientific notation.
Scientific Notation
A positive number is written in scientific notation when it is expressed in the form
Where a is a number greater than or equal to 1 and less than 10 and n is an integer.
10 ,na
(1 10)a
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32
Scientific Notation
Converting from Scientific to Decimal Notation
We can use n, the exponent on the 10 in , to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left places.
10na
n
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33
Objective #3: Example
3a. Write 97.4 10 in decimal notation. The exponent is positive so we move the decimal point nine places to the right.
97.4 10 7,400,000,000
3b. Write 63.017 10 in decimal notation. The exponent is negative so we move the decimal point six places to the left.
63.017 10 0.000003017
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34
Objective #3: Example
3a. Write 97.4 10 in decimal notation. The exponent is positive so we move the decimal point nine places to the right.
97.4 10 7,400,000,000
3b. Write 63.017 10 in decimal notation. The exponent is negative so we move the decimal point six places to the left.
63.017 10 0.000003017
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35
Objective #4 Convert from decimal notation to
scientific notation.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36
Converting from Decimal to Scientific Notation
• Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10.
• Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal was moved. The exponent n is positive if the given number is great than 10 and negative if the given number is between 0 and 1.
Scientific Notation
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 37
Objective #4: Example
4a. Write 7,410,000,000 in scientific notation.
97,410,000,000 7.41 10
4b. Write 0.000000092 in scientific notation.
80.000000092 9.2 10
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 38
Objective #4: Example
4a. Write 7,410,000,000 in scientific notation.
97,410,000,000 7.41 10
4b. Write 0.000000092 in scientific notation.
80.000000092 9.2 10
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Objective #5 Compute with scientific notation.
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Computations with Scientific Notation
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Computations with Scientific Notation
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Perform the indicated computation, writing the answer in scientific notation.
86 102.31041.5
13107312.1
6 85.41 3.2 10 10
86 102.31041.5
Regroup factors
Multiply
Simplify
Rewrite in scientific notation
1410312.17 114107312.1
8610312.17
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 42
Computations with Scientific Notation
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Perform the indicated computation, writing the answer in scientific notation.
6
8
1041.5
102.3
Group factors
Divide
Simplify
Write in scientific notation
681059.0 21059.0
3109.5
6
8
10
10
41.5
2.3
6
8
1041.5
102.3
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 43
Objective #5: Example
5a. Perform the indicated computation, writing the answer in
scientific notation: 8 2(3 10 )(2 10 )
8 2 8 2
8 2
10
(3 10 )(2 10 ) (3 2) (10 10 )
6 10
6 10
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Objective #5: Example
5a. Perform the indicated computation, writing the answer in
scientific notation: 8 2(3 10 )(2 10 )
8 2 8 2
8 2
10
(3 10 )(2 10 ) (3 2) (10 10 )
6 10
6 10
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Objective #5: Example
5b. Perform the indicated computation, writing the answer in
scientific notation: 7
48.4 10
4 10
7 7
4 4
7 ( 4)
11
8.4 10 8.4 1044 10 10
2.1 10
2.1 10
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Objective #5: Example
5b. Perform the indicated computation, writing the answer in
scientific notation: 7
48.4 10
4 10
7 7
4 4
7 ( 4)
11
8.4 10 8.4 1044 10 10
2.1 10
2.1 10
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5c. Perform the indicated computation, writing the answer in
scientific notation: 2 3(4 10 )
2 3 3 2 3
6
5
(4 10 ) 4 (10 )
64 10
6.4 10
Objective #5: Example
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5c. Perform the indicated computation, writing the answer in
scientific notation: 2 3(4 10 )
2 3 3 2 3
6
5
(4 10 ) 4 (10 )
64 10
6.4 10
Objective #5: Example
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Objective #6 Solve applied problems using scientific notation.
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Scientific Notation
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia?
6102.7
We will first write 7.2 million in scientific notation. It is:
81066.3
Now we can answer the question.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 51
Scientific Notation
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia?
6102.7
We will first write 7.2 million in scientific notation. It is:
81066.3
Now we can answer the question.
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Scientific Notation
Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.
CONTINUEDCONTINUED
Divide ‘dollar amount’ by ‘acreage’
Divide
Simplify8
6
10
10
66.3
2.7
6
8
7.2 10
3.66 10
21097.1
861097.1 Subtract
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 53
Scientific Notation
Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.
CONTINUEDCONTINUED
Divide ‘dollar amount’ by ‘acreage’
Divide
Simplify8
6
10
10
66.3
2.7
6
8
7.2 10
3.66 10
21097.1
861097.1 Subtract
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 54
Objective #6: Example
6. The cost of President Obama’s 2009 economic stimulus package was $787 billion, or 117.87 10 dollars. If this cost were evenly divided among every individual in the United States (approximately 83.07 10 people), how much would each citizen have to pay?
11 11
38 8
7.87 10 7.87 102.56 10 2560
3.073.07 10 10
Each citizen would have to pay about $2560.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 55
Objective #6: Example
6. The cost of President Obama’s 2009 economic stimulus package was $787 billion, or 117.87 10 dollars. If this cost were evenly divided among every individual in the United States (approximately 83.07 10 people), how much would each citizen have to pay?
11 11
38 8
7.87 10 7.87 102.56 10 2560
3.073.07 10 10
Each citizen would have to pay about $2560.